The Mutual-Visibility Problem In Directed Graphs

TL;DR

This paper extends the mutual-visibility concept from undirected to directed graphs, establishing fundamental results for various classes, analyzing algorithmic complexity, and exploring structural impacts on mutual-visibility numbers.

Contribution

It provides the first comprehensive analysis of mutual-visibility in directed graphs, including exact values for DAGs, cycles, and tournaments, and proves NP-hardness for general digraphs.

Findings

Mutual-visibility number is 1 for DAGs.

Mutual-visibility number is 2 for directed cycles of length ≥3.

Tournaments can have arbitrarily large mutual-visibility sets, growing linearly with size.

Abstract

The study of mutual visibility has traditionally focused on undirected graphs, asking for the maximum number of vertices that can communicate via shortest paths without intermediate interference from other set members. In this paper, we extend this concept to directed graphs, establishing fundamental results for several graph classes. We prove that for Directed Acyclic Graphs (DAGs), the mutual-visibility number $\mu(D)$ is always 1, and for directed cycles of length $n\ge3$, it is strictly 2. In contrast, we demonstrate that tournaments can support arbitrarily large mutual-visibility sets; specifically, using properties of Paley tournaments, we show that $\mu(T)$ grows linearly with the size of the tournament. On the algorithmic side, we show that while verifying a candidate set is polynomial-time solvable ($O(|S|(|V|+|A|))$), the problem of determining $\mu(D)$ is NP-hard for general digraphs. We also analyze the impact of strong bridges and strongly connected components on the upper bounds of $\mu(D)$.